Microlocal resolvent estimates, revisited

TL;DR

This paper refines microlocal resolvent estimates for Schrödinger operators, extending results to broader classes of pseudodifferential operators on manifolds, with proofs based on propagation estimates.

Contribution

It provides a refined analysis of the wave front set of the resolvent kernel for Schrödinger-type operators, applicable to discrete and higher-order operators, using propagation estimates.

Findings

Refined microlocal resolvent estimates for Schrödinger operators.

Extension to pseudodifferential operators on manifolds.

Applicable to discrete Schrödinger and higher-order operators.

Abstract

Let $H$ be a Schr\"odinger type operator with long-range perturbation. We study the wave front set of the distribution kernel of $(H-\lambda\mp i0)^{-1}$, where $\lambda$ is in the absolutely continous spectrumof $H$.The result is a refinement of the microlocal resolvent estimate of Isozaki-Kitada \cite{IK1,IK2}. We prove the result for a class of pseudodifferential operators on manifolds so that they apply to discrete Schr\"odinger operators and higher order operators on the Euclidean space. The proof relies on propagation estimates, whereas the original proof of Isozaki-Kitada relies on a construction of parametrices.